1. Field of the Invention
The present invention relates to electromagnetic field simulation technology used in developing and designing high-frequency products such as multilayer products, giga filters, and EMI (Electro-Magnetic Interference) elimination filters, and more particularly, the present invention relates to an electromagnetic field analyzer, an electromagnetic field analyzing program, and a storage medium for recording the program in which the simulation time is reduced.
2. Description of the Related Art
In recent years, the development and design of high-frequency products have been aggressively pursued, and, in order to perform an electromagnetic field analysis of high-frequency products, electromagnetic field simulation software has been commonly used. Generally, there are many instances where a finite element method is used for electromagnetic field analysis. In the electromagnetic field simulation using the finite element method, since it takes a long time to solve simultaneous linear equations, high-speed solving methods are desired. In particular, in the field of electromagnetic field analysis, since only a direct method called a Gauss' elimination method can be used, the problem of long calculation times is more serious.
However, a paper (R. Hiptmair, “Multigrid Method for Maxwell's Equations,” SIAM Journal of Numerical Analysis, vol. 36, no. 1, pp. 204–225, 1999), which is referred to herein as Non-patent document 1, and which was made public by Hiptmair in 1999, proved that a solution method called a multi-grid method can be also used in an electromagnetic field analysis and the method greatly reduces the calculation time as compared to a direct method.
In the multi-grid method, an analysis object is divided into two classes of fine elements and coarse elements, and the solution of fine elements is determined using the solution of coarse elements.
When simultaneous linear equations are calculated by the Gauss' elimination method, generally the calculation time is proportional to the third power of the dimensions of a matrix. Since the size of the dimensions corresponds to the number of elements, when the number of elements is doubled, the calculation time is 8 times as large, and, when the number of elements is 10 times as large, the calculation time is 1,000 times as large. When the Gauss' direct method is applied to only coarse elements, an approximate solution to fine elements is obtained by using the solution to coarse elements, and the Gauss' elimination method is not used, the calculation time for fine elements can be reduced to ⅛ and 1/1000, respectively. Since additional calculation time is added, although the practical calculation time is not reduced to such a great extent, the calculation time is always greatly reduced.
The multi-grid method can be broken down into two methods: a method using a nested mesh and a method using a non-nested mesh. However, since a prolongation matrix is incomplete as described later, the method using a nested mesh must be used.
FIGS. 12A and 12B show the division into elements using a nested mesh. FIG. 12A shows the division into elements using a nested mesh in the case of two-dimensional analysis where a coarse triangular element is uniformly divided to form fine triangular elements. Furthermore, FIG. 12B shows the division into elements using a nested mesh in the case of three-dimensional analysis where a coarse tetrahedral element is uniformly divided to form eight fine tetrahedral elements. As understood from FIGS. 12A and 12B, there are geometrical restrictions between coarse elements and fine elements when using a nested mesh.
Another prior art method is described in non-patent document 2: D. Dibben and T. Yamada, “Non-Nested Multigrid and Automatic Mesh Coarsening for High Frequency Electromagnetic Problems” IEEJ Investigating Research Committee Material, SA-02-34, pp. 71–75, 2002.
FIGS. 13A and 13B describe a first problem that occurs when using a nested mesh. FIG. 13A shows how a circle is divided into elements using a nested mesh to form coarse elements. Furthermore, FIG. 13B shows how coarse elements are divided into fine elements using a nested mesh. As understood from FIGS. 13A and 13B, a curved surface cannot be correctly expressed because of the geometrical restrictions of a nested mesh.
Furthermore, FIG. 13C shows how a circle is divided into elements using a non-nested mesh to form fine elements. As understood from FIG. 13C, since there are no geometrical restrictions when using a non-nested mesh, a curved surface can be correctly expressed.
FIGS. 14A to 14C describe a second problem that occurs when using a nested mesh. FIG. 14A shows how a square is divided into elements using a nested mesh to form coarse elements. Furthermore, FIG. 14B shows how coarse elements are divided into fine elements. As understood from FIGS. 14A and 14B, coarse elements cannot be partially divided into fine elements because of the geometrical restrictions of a nested mesh.
Furthermore, FIG. 14C shows how a square is divided into elements using a non-nested mesh to form fine elements. As understood from FIG. 14C, coarse elements can be partially divided into fine elements because there are no geometrical restrictions when using a non-nested mesh.
Most industrial products have curved surfaces, such as a circular cylinder and square surfaces, and they inevitably contain partially fine portions. Accordingly, it is difficult to apply a nested mesh to industrial products. On the other hand, since there are no geometrical restrictions in a non-nested mesh, fine elements can be freely formed. However, since the positional relationship is not systematic, it is difficult to make the electromagnetic field of coarse elements related to the electromagnetic field of fine elements. It is a prolongation matrix to make coarse elements related to fine elements, but no precise prolongation matrix has been found. This means that a multi-grid method cannot be practically used, and accordingly, nothing can be used except for a Gauss' direct method, which requires a lot of time, in the analysis of magnetic field.